A water tank is shaped like an inverted right circular cone with a height of 10 m and…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A water tank is shaped like an inverted right circular cone with a height of 10 m and a top radius of 5 m. Water is being pumped into the tank at a rate of 3 m

^{3}

/min while a small hole at the vertex allows water to leak out at a rate of 1 m

^{3}

/min. Assume that at time

t=0

the tank is empty. (a) Express the radius

r

of the water’s surface as a function of the water depth

h

using geometric considerations. (b) Using your result from part (a), write the volume

V

of water in the tank solely as a function of

h

. (c) Derive an explicit formula for

\dfrac{dh}{dt}

in terms of

h

, given the net rate

\dfrac{dV}{dt}

of change of volume in the tank. (d) Calculate

\dfrac{dh}{dt}

when the water depth is 4 m and interpret the meaning of its sign in the context of the problem.

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